Distributed Approximation of Capacitated Dominating Sets

We study local, distributed algorithms for the capacitated minimum dominating set (CapMDS) problem, which arises in various distributed network applications. Given a network graph G=(V,E), and a capacity cap(v)∈ℕ for each node v∈V, the CapMDS problem asks for a subset S⊆V of minimal cardinality, such that every network node not in S is covered by at least one neighbor in S, and every node v∈S covers at most cap(v) of its neighbors. We prove that in general graphs and even with uniform capacities, the problem is inherently non-local, i.e., every distributed algorithm achieving a non-trivial approximation ratio must have a time complexity that essentially grows linearly with the network diameter. On the other hand, if for some parameter ε>0, capacities can be violated by a factor of 1+ε, CapMDS becomes much more local. Particularly, based on a novel distributed randomized rounding technique, we present a distributed bi-criteria algorithm that achieves an O(log Δ)-approximation in time O(log 3n+log (n)/ε), where n and Δ denote the number of nodes and the maximal degree in G, respectively. Finally, we prove that in geometric network graphs typically arising in wireless settings, the uniform problem can be approximated within a constant factor in logarithmic time, whereas the non-uniform problem remains entirely non-local.

[1]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[2]  Roger Wattenhofer,et al.  Fast Deterministic Distributed Maximal Independent Set Computation on Growth-Bounded Graphs , 2005, DISC.

[3]  Michael Elkin,et al.  Distributed approximation: a survey , 2004, SIGA.

[4]  Nathan Linial,et al.  Locality in Distributed Graph Algorithms , 1992, SIAM J. Comput..

[5]  Fabrizio Grandoni,et al.  Primal-dual based distributed algorithms for vertex cover with semi-hard capacities , 2005, PODC '05.

[6]  Peng-Jun Wan,et al.  Distributed Construction of Connected Dominating Set in Wireless Ad Hoc Networks , 2002, Proceedings.Twenty-First Annual Joint Conference of the IEEE Computer and Communications Societies.

[7]  Judit Bar-Ilan,et al.  Generalized submodular cover problems and applications , 2001, Theor. Comput. Sci..

[8]  Moni Naor,et al.  What can be computed locally? , 1993, STOC.

[9]  Roger Wattenhofer,et al.  On the locality of bounded growth , 2005, PODC '05.

[10]  Judit Bar-Ilan,et al.  How to Allocate Network Centers , 1993, J. Algorithms.

[11]  Thomas Moscibroda,et al.  What Cannot Be Computed Locally , 2004 .

[12]  Shay Kutten,et al.  Fast Distributed Construction of Small k-Dominating Sets and Applications , 1998, J. Algorithms.

[13]  Roger Wattenhofer,et al.  Maximal independent sets in radio networks , 2005, PODC '05.

[14]  Xiang-Yang Li,et al.  Distributed low-cost backbone formation for wireless ad hoc networks , 2005, MobiHoc '05.

[15]  Beat Gfeller,et al.  A randomized distributed algorithm for the maximal independent set problem in growth-bounded graphs , 2007, PODC '07.

[16]  Wendi Heinzelman,et al.  Energy-efficient communication protocol for wireless microsensor networks , 2000, Proceedings of the 33rd Annual Hawaii International Conference on System Sciences.

[17]  Peng-Jun Wan,et al.  Message-optimal connected dominating sets in mobile ad hoc networks , 2002, MobiHoc '02.

[18]  Roger Wattenhofer,et al.  A log-star distributed maximal independent set algorithm for growth-bounded graphs , 2008, PODC '08.

[19]  B. R. Badrinath,et al.  On the node-scheduling approach to topology control in ad hoc networks , 2005, MobiHoc '05.

[20]  Baruch Awerbuch,et al.  Complexity of network synchronization , 1985, JACM.

[21]  Adrian Perrig,et al.  Using Clustering Information for Sensor Network Localization , 2005, DCOSS.

[22]  Michael Luby,et al.  A simple parallel algorithm for the maximal independent set problem , 1985, STOC '85.

[23]  Leonidas J. Guibas,et al.  Discrete mobile centers , 2001, SCG '01.

[24]  Rajiv Gandhi,et al.  Distributed Algorithms for Coloring and Domination in Wireless Ad Hoc Networks , 2004, FSTTCS.

[25]  Jie Wu,et al.  An extended localized algorithm for connected dominating set formation in ad hoc wireless networks , 2004, IEEE Transactions on Parallel and Distributed Systems.

[26]  Noga Alon,et al.  A Fast and Simple Randomized Parallel Algorithm for the Maximal Independent Set Problem , 1985, J. Algorithms.

[27]  Christoph Lenzen,et al.  Leveraging Linial's Locality Limit , 2008, DISC.

[28]  N Linial,et al.  Low diameter graph decompositions , 1993, Comb..

[29]  Idit Keidar,et al.  Veracity radius: capturing the locality of distributed computations , 2006, PODC '06.

[30]  Roger Wattenhofer,et al.  The price of being near-sighted , 2006, SODA '06.

[31]  Prabhakar Raghavan,et al.  Randomized rounding: A technique for provably good algorithms and algorithmic proofs , 1985, Comb..

[32]  Roger Wattenhofer,et al.  Constant-time distributed dominating set approximation , 2003, PODC '03.

[33]  R. Rajaraman,et al.  An efficient distributed algorithm for constructing small dominating sets , 2002 .